Solve each equation using the square root property. (4x + 1)² = 20

Verified step by step guidance

1. Start with the given equation: \( (4x + 1)^2 = 20 \).
2. Apply the square root property, which states that if \( A^2 = B \), then \( A = \pm \sqrt{B} \). So, take the square root of both sides: \( 4x + 1 = \pm \sqrt{20} \).
3. Simplify the square root if possible. Since \( \sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5} \), rewrite the equation as \( 4x + 1 = \pm 2\sqrt{5} \).
4. Isolate the variable term by subtracting 1 from both sides: \( 4x = -1 \pm 2\sqrt{5} \).
5. Solve for \( x \) by dividing both sides by 4: \( x = \frac{-1 \pm 2\sqrt{5}}{4} \).

Key Concepts

Square Root Property

The square root property states that if an equation is in the form (ax + b)^2 = c, then ax + b = ±√c. This allows you to solve quadratic equations by isolating the squared term and taking the square root of both sides, considering both positive and negative roots.

Isolating the Variable

Before applying the square root property, you must isolate the squared expression on one side of the equation. This involves algebraic manipulation such as addition, subtraction, multiplication, or division to simplify the equation and prepare it for taking square roots.

Solving Linear Equations

After applying the square root property, you get two linear equations to solve for the variable. Solving these involves standard algebraic steps like adding or subtracting constants and dividing by coefficients to find the variable's values.